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Is |x| = y - z ? (1) x + y = z (2) x < 0

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DenisSh
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Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   Updated on: 01 Jun 2021, 03:27
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Is |x| = y - z ?

(1) x + y = z
(2) x < 0
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Official Answer
C

Originally posted by DenisSh on 04 Oct 2009, 23:26.
Last edited by Bunuel on 01 Jun 2021, 03:27, edited 3 times in total.
Edited the question and added the OA
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Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   05 Oct 2009, 05:31
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Is |x| = y - z[/m]?

For a start, take into account that for the equation |x| = y - z to hold true, y - z must be greater than or equal to 0. This is due to the fact that it's equated to the absolute value of a number, |x|, which cannot be negative. In essence, the question is asking whether y - z is non-negative and if their difference equals |x|.

(1) -x = y - z.

If x > 0, -x would be negative, which would imply that y - z is also negative. This doesn't meet our initial criteria. For instance, if x = 1 and y - z = -1, then (|x| = 1) is not equal to (y - z = -1).

If x ≤ 0, -x would be non-negative, which would imply y - z is also non-negative, a valid scenario. For instance, if x = -1 and y - z = 1, then (|x| = 1) is equal to (y - z = 1).

This gives us two possible outcomes and does not definitively answer our question. Not sufficient.

(2) \(x<0\).

This condition alone is not sufficient because it only gives information about x. We still need to know whether the value of y - z equals |x|.

(1) + (2) Given that x < 0, |x| = -x, and so our original question becomes: is -x = y - z? The second statement gives a direct YES answer to this question. Sufficient.

Answer: C.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   27 Jun 2011, 21:32
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akhileshgupta05 wrote:
is \(|x|=y-z\) ?

(1)\(x+y=z\)
(2)\(x<0\)

Please give detailed explanation for your answer.


|X|= y-z means;
X>0; X= Y-Z
X<0; -X= Y-Z ; X= Z-Y

St1: X= Z-Y We dont know if X>0 or <0 ............ insufficient

St2: X<0 ; when dont know about Y and Z ......... insufficient

1&2 : we know X<0 and X= Z-Y hence sufficient
General Discussion
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   05 Oct 2009, 05:58
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Bunuel wrote:
y-z=|x|? --> y-z must be >=0...


Brilliant, thank you! :^)
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   01 Aug 2010, 14:15
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is |x|=y-z?

given:
(1) x+y=z
(2) x<0

solving (1) first:

y=z-x
|x|=(z-x)-z
|x|=-x

take x=1, z=2, y=1
1=1-2 (no)
take x=-1, z=2, y=3
|x|=y-z?
|-1|=3-2=1 YES

so what solving for |x|=-x meant was that x MUST be negative for the equation to be true, if it is positive then it is not true (since in that case, |x| would not equal -x).

hence the answer is C.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   28 Jul 2012, 13:50
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sayak636 wrote:
Is |X|= Y- Z?

1. X+Y= Z
2. X< 0


(1) Can be rewritten as X = -Y + Z, so |X| = |-Y + Z|, which would be equal to Y - Z, if and only if \(-Y+Z\leq0\). Obviously, we don't know that, so (1) insufficient.
(2) Cannot be sufficient, it doesn't say anything about Y and Z.
(1) and (2) together: X = -Y + Z < 0, therefore |X| = Y - Z, sufficient.

Answer: C
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   05 Oct 2012, 05:44
Bunuel wrote:
Is \(|x|=y-z\)?

Note that \(y-z\) must be \(\geq{0}\), because absolute value (in our case \(|x|\)) can not be negative.

Generally question asks whether \(y-z\geq{0}\) and whether the difference between them equals to \(|x|\).

(1) \(-x=y-z\)
if \(x>0\) --> \(y-z\) is negative --> no good for us;
if \(x\leq{0}\) --> \(y-z\) is positive --> good.
Two possible answers not sufficient;

(2) \(x<0\)
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.



Hi bunuel,
I am not able to understand the solution for this problem. Can you kindly explain the highlighted areas.
Note that y-z must be \geq{0}, because absolute value (in our case |x|) can not be negative.

Generally question asks whether y-z\geq{0} and whether the difference between them equals to |x|.

(1) -x=y-z
if x>0 --> y-z is negative --> no good for us;
if x\leq{0} --> y-z is positive --> good.

Waiting for reply.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   05 Oct 2012, 06:12
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fameatop wrote:
Bunuel wrote:
Is \(|x|=y-z\)?

Note that \(y-z\) must be \(\geq{0}\), because absolute value (in our case \(|x|\)) can not be negative.

Generally question asks whether \(y-z\geq{0}\) and whether the difference between them equals to \(|x|\).

(1) \(-x=y-z\)
if \(x>0\) --> \(y-z\) is negative --> no good for us;
if \(x\leq{0}\) --> \(y-z\) is positive --> good.
Two possible answers not sufficient;

(2) \(x<0\)
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.



Hi bunuel,
I am not able to understand the solution for this problem. Can you kindly explain the highlighted areas.
Note that y-z must be \geq{0}, because absolute value (in our case |x|) can not be negative.

Generally question asks whether y-z\geq{0} and whether the difference between them equals to |x|.

(1) -x=y-z
if x>0 --> y-z is negative --> no good for us;
if x\leq{0} --> y-z is positive --> good.

Waiting for reply.


Look at \(|x|=y-z\): the left hand side is absolute value (|x|), which cannot be negative, hence the right hand side (y-z) also cannot be negative. Therefore must be true that \(y-z\geq{0}\).

Next, for (1) given that \(-x=y-z\). Now, if \(x>0\), or if \(x\) is positive, then we'll have that \(-positive =y-z\) --> \(negative=y-z\). But as we concluded above \(y-z\) cannot be negative, hence this scenario is not good.

Hope it's clear.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   20 Oct 2016, 11:29
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yashrakhiani wrote:
IS |x| = y -z??

1)x+ y = z


2)x<0


Answer is C.

From the question,
if x>0, x = y-z
if x<0, x = -(y-z) = z-y

Let's start with the easy statement - Statement 2.

(2): Insufficient
x<0 doesn't give us any information about y and z. Eliminate B and D

(1): Insufficient
Statement 1 says that x = z - y
If x<0 , then the answer is yes since |x| will be equal to y-z
if x>0, then the answer is no since |x| = z-y
We have a Yes and a No. Eliminate A.

(1) and (2) together: Sufficient
We have x<0 and x = z - y. hence, |x| = y-z

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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   13 Jul 2019, 21:13
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|x| = y-z? ==> y-z>0? (since |x| > 0) ==> y>z?

St 1: x + y = z ==> Not sufficient since we do not know the signs

St 2: x<0 ==> Not sufficient since we are not told anything about z and y

Sts 1 and 2 together ==> From St 1 we see that x= z-y and from St 2 x<0 ==> z-y<0 ==> z<y ==> sufficient since it answers the question. Therefore the answer is C
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   17 Aug 2019, 07:21
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|x| is always positive so the question asks us is the (y-z) a positive magnitude of X.
1. Now with stm-1 y-z=-x , this could be positive value of x or negative value of x depending on the value of x. Yes and No, so Not sufficient.
2. Stm -2 X< 0, what about the remaining terms - not sufficient.

1+2, X<0 so -X>0, so (y-z)= positive x = |x|, which is always positive.

Ans- C
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   13 May 2023, 01:07
Bunuel wrote:
Is \(|x|=y-z\)?

Note that \(y-z\) must be \(\geq{0}\), because absolute value (in our case \(|x|\)) can not be negative.

Generally question asks whether \(y-z\geq{0}\) and whether the difference between them equals to \(|x|\).

(1) \(-x=y-z\)
if \(x>0\) --> \(y-z\) is negative --> no good for us;
if \(x\leq{0}\) --> \(y-z\) is positive --> good.
Two possible answers not sufficient;

(2) \(x<0\)
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.


Bunuel KarishmaB
I solved it this way. Let me know if there are any conceptual gaps in this method:
Qs: |x| = y-z
Thus qs is : Is -x=z-y or x=y-z

St 1:
x+y=z
x=z-y
This is neither of the equations: -x=z-y or x=y-z
Not sufficient

St 2:
Clearly Insufficient

St: 1+2
x is negative. Substituting x=-x in x=z-y
-x=z-y
One of the derived question equations
Thus sufficient

I am a bit unsure about the substitution. Looking forward to your response for more clarity.
Thanks!
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   13 May 2023, 20:59
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RenB wrote:
Bunuel wrote:
Is \(|x|=y-z\)?

Note that \(y-z\) must be \(\geq{0}\), because absolute value (in our case \(|x|\)) can not be negative.

Generally question asks whether \(y-z\geq{0}\) and whether the difference between them equals to \(|x|\).

(1) \(-x=y-z\)
if \(x>0\) --> \(y-z\) is negative --> no good for us;
if \(x\leq{0}\) --> \(y-z\) is positive --> good.
Two possible answers not sufficient;

(2) \(x<0\)
Not sufficient (we need to know value of y-z is equal or not to |x|)

(1)+(2) Sufficient.

Answer: C.


Bunuel KarishmaB
I solved it this way. Let me know if there are any conceptual gaps in this method:
Qs: |x| = y-z
Thus qs is : Is -x=z-y or x=y-z



Not Correct.
Question: Is |x| = y - z
means
Is x = y - z if x is positive or is -x = y - z if x is negative.

We use the definition of absolute values to replace |x| by some expression in x. There is no multiplication by -1 on both sides.
I suggest you to check out the absolute values section of some test prep company content.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink] New post   01 Aug 2023, 10:54
We have :
Is |x| = y - z ?

(1) x + y = z
(2) x < 0

if x >0 then inequality will be x= y-z ; if x < 0 then inequality will be x= z-y
1. x= z-y we can not say for sure if the condition hold
2. x <0 then again we can not sure for sure
combining both x < 0 then x = z-y which is what we are getting above in the inequality.
Hence C is the correct option.
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Re: Is |x| = y - z ? (1) x + y = z (2) x < 0 [#permalink]
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